Algebraic Expressions and Identities - Class 8 Mathematics - Chapter 6 - Notes, NCERT Solutions & Extra Questions
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Exercise 6.1 - Algebraic Expressions and Identities | R.D. Sharma | Mathematics | Class 8
Identify the terms, their coefficients for each of the following expressions:
(i) $7 x^{2} y z-5 x y$
(ii) $x^{2}+x+1$
(iii) $3 x^{2} y^{2}-5 x^{2} y^{2} z^{2}+z^{2}$
(iv) $9-a b+b c-c a$
(v) $\frac{a}{2}+\frac{b}{2}-a b$
(vi) $0.2 x-0.3 x y+0.5 y$
For each expression, we'll identify the terms and then the coefficients for each term:
(i) Expression: $7 x^2 y z - 5 x y$
Terms: $7 x^2 y z$, $-5 x y$
Coefficients: $7$ for $x^2 y z$, $-5$ for $x y$
(ii) Expression: $x^2 + x + 1$
Terms: $x^2$, $x$, $1$
Coefficients: $1$ for $x^2$, $1$ for $x$, $1$ for the constant term
(iii) Expression: $3 x^2 y^2 - 5 x^2 y^2 z^2 + z^2$
Terms: $3 x^2 y^2$, $-5 x^2 y^2 z^2$, $z^2$
Coefficients: $3$ for $x^2 y^2$, $-5$ for $x^2 y^2 z^2$, $1$ for $z^2$
(iv) Expression: $9 - a b + b c - c a$
Terms: $9$, $-a b$, $b c$, $-c a$
Coefficients: $1$ for the constant term $9$, $-1$ for $a b$, $1$ for $b c$, $-1$ for $c a$
(v) Expression: $\frac{a}{2} + \frac{b}{2} - a b$
Terms: $\frac{a}{2}$, $\frac{b}{2}$, $-a b$
Coefficients: $\frac{1}{2}$ for $a$, $\frac{1}{2}$ for $b$, $-1$ for $ab$
(vi) Expression: $0.2 x - 0.3 x y + 0.5 y$
Terms: $0.2 x$, $-0.3 x y$, $0.5 y$
Coefficients: $0.2$ for $x$, $-0.3$ for $x y$, $0.5$ for $y$
Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category?
(i) $x+y$
(ii) 1000
(iv) $7+a+5 b$
(v) $2 b-3 b^{2}$
(vii) $5 x-4 y+3 x$
(viii) $4 a-15 a^{2}$
(x) $p q r$
(xi) $p^{2} q+p q^{2}$
(iii) $x+x^{2}+x^{3}+x^{4}$
(vi) $2 y-3 y^{2}+4 y^{3}$
(ix) $x y+y z+z t+t x$
(xii) $2 p+2 q$
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Ask Chatterbot AIExercise 6.2 - Algebraic Expressions and Identities | R.D. Sharma | Mathematics | Class 8
Add the following algebraic expressions:
(i) $3 a^{2} b,-4 a^{2} b, 9 a^{2} b$
(ii) $\frac{2}{3} a, \frac{3}{5} a,-\frac{6}{5} a$
(iii) $4 x y^{2}-7 x^{2} y, 12 x^{2} y-6 x y^{2},-3 x^{2} y+5 x y^{2}$
(iv) $\frac{3}{2} a-\frac{5}{4} b+\frac{2}{5} c, \frac{2}{3} a-\frac{7}{2} b+\frac{7}{2} c, \frac{5}{3} a+\frac{5}{2} b-\frac{5}{4} c$
(v) $\frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x,-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y$
(vi) $\frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}, \frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3}, \frac{3}{2} x^{2}-\frac{5}{2} x-2$
The results for the addition of the given algebraic expressions are:
(i) $8 a^{2} b$
(ii) $\frac{a}{15}$
(iii) $2 x^2 y + 3 x y^2$
(iv) $\frac{23}{6} a - \frac{9}{4} b + \frac{53}{20} c$
(v) $\frac{51}{14} x y - \frac{19}{35} x - \frac{31}{10} y$
(vi) $5 x^3 + \frac{11}{4} x^2 - \frac{7}{2} x$
These results are simplified forms of the expressions after addition.
Subtract:
(i) $-5 x y$ from $12 x y$
(ii) $2 a^{2}$ from $-7 a^{2}$
(iii) $2 a-b$ from $3 a-5 b$
(iv) $2 x^{3}-4 x^{2}+3 x+5$ from $4 x^{3}+x^{2}+x+6$
(v) $\frac{2}{3} y^{3}-\frac{2}{7} y^{2}-5$ from $\frac{1}{3} y^{3}+\frac{5}{7} y^{2}+y-2$
(vi) $\frac{3}{2} x-\frac{5}{4} y-\frac{7}{2} z$ from $\frac{2}{3} x+\frac{3}{2} y-\frac{4}{3} z$
(vii) $x^{2} y-\frac{4}{5} x y^{2}+\frac{4}{3} x y$ from $\frac{2}{3} x^{2} y+\frac{3}{2} x y^{2}-\frac{1}{3} x y$
(viii) $\frac{a b}{7}-\frac{35}{3} b c+\frac{6}{5} a c$ from $\frac{3}{5} b c-\frac{4}{5} a c$
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Take away:
(i) $\frac{6}{5} x^{2}-\frac{4}{5} x^{3}+\frac{5}{6}+\frac{3}{2} x$ from $\frac{x^{3}}{3}-\frac{5}{2} x^{2}+\frac{3}{5} x+\frac{1}{4}$
(ii) $\frac{5 a^{2}}{2}+\frac{3 a^{3}}{2}+\frac{a}{3}-\frac{6}{5}$ from $\frac{1}{3} a^{3}-\frac{3}{4} a^{2}-\frac{5}{2}$
(iii) $\frac{7}{4} x^{3}+\frac{3}{5} x^{2}+\frac{1}{2} x+\frac{9}{2}$ from $\frac{7}{2}-\frac{x}{3}-\frac{x^{2}}{5}$
(iv) $\frac{y^{3}}{3}+\frac{7}{3} y^{2}+\frac{1}{2} y+\frac{1}{2}$ from $\frac{1}{3}-\frac{5}{3} y^{2}$
(v) $\frac{2}{3} a c-\frac{5}{7} a b+\frac{2}{3} b c$ from $\frac{3}{2} a b-\frac{7}{4} a c-\frac{5}{6} b c$
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Subtract $3 x-4 y-7 z$ from the sum of $x-3 y+2 z$ and $-4 x+9 y-11 z$.
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Subtract the sum of $3 l-4 m-7 n^{2}$ and $2 l+3 m-4 n^{2}$ from the sum of $9 l+2 m-3 n^{2}$ and $-3 l+m+4 n^{2} \ldots .$.
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Subtract the sum of $2 x-x^{2}+5$ and $-4 x-3+7 x^{2}$ from 5 .
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Simplify each of the following:
(i) $x^{2}-3 x+5-\frac{1}{2}\left(3 x^{2}-5 x+7\right)$
(ii) $[5-3 x+2 y-(2 x-y)]-(3 x-7 y+9)$
(iii) $\frac{11}{2} x^{2} y-\frac{9}{4} x y^{2}+\frac{1}{4} x y-\frac{1}{14} y^{2} x+\frac{1}{15} y x^{2}+\frac{1}{2} x y$
(iv) $\left(\frac{1}{3} y^{2}-\frac{4}{7} y+11\right)-\left(\frac{1}{7} y-3+2 y^{2}\right)-\left(\frac{2}{7} y-\frac{2}{3} y^{2}+2\right)$
(v) $-\frac{1}{2} a^{2} b^{2} c+\frac{1}{3} a b^{2} c-\frac{1}{4} a b c^{2}-\frac{1}{5} c b^{2} a^{2}+\frac{1}{6} c b^{2} a-\frac{1}{7} c^{2} a b+\frac{1}{8} c a^{2} b$.
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Ask Chatterbot AIExercise 6.3 - Algebraic Expressions and Identities | R.D. Sharma | Mathematics | Class 8
Find each of the following products:
$5 x^{2} \times 4 x^{3}$
The product of the expressions $5 x^2$ and $4 x^3$ is $20 x^5$.
Find each of the following products:
$-3 a^{2} \times 4 b^{4}$
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Find each of the following products:
$(-5 x y) \times\left(-3 x^{2} y z\right)$
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Find each of the following products:
$\frac{1}{2} x y \times \frac{2}{3} x^{2} y z^{2}$
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Find each of the following products:
$\left(-\frac{7}{5} x y^{2} z\right) \times\left(\frac{13}{3} x^{2} y z^{2}\right)$
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Find each of the following products:
$\left(\frac{-24}{25} x^{3} z\right) \times\left(-16 x z^{9} y\right)$
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Find each of the following products:
$\left(-\frac{1}{27} a^{2} b^{2}\right) \times\left(\frac{9}{2} a^{3} b^{2} c^{2}\right)$
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Find each of the following products:
$(-7 x y) \times\left(\frac{1}{4} x^{2} y\right)$
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Find each of the following products:
$$ (7 a b) \times\left(-5 a b^{2} c\right) \times\left(6 a b c^{2}\right) $$
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Find each of the following products:
$(-5 a) \times\left(-10 a^{4}\right) \times\left(-2 a^{3}\right)$
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Find each of the following products:
$\left(-4 x^{2}\right) \times\left(-6 x y^{2}\right) \times\left(-3 y z^{2}\right)$
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Find each of the following products:
$\left(-\frac{2}{7} a^{4}\right) \times\left(-\frac{3}{4} a^{2} b\right) \times\left(-\frac{14}{5} b^{3}\right)$
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Find each of the following products:
$\left(\frac{7}{9} a b^{2}\right) \times\left(\frac{15}{7} a c^{2} b\right) \times\left(-\frac{3}{5} a^{2} c\right)$
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Find each of the following products:
$\left(\frac{4}{3} u^{2} v w\right) \times\left(-5 u v w^{2}\right) \times\left(\frac{1}{3} v^{2} w u\right)$
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Find each of the following products:
$(0.5 x) \times\left(\frac{1}{3} x y^{2} z^{4}\right) \times\left(24 x^{2} y z\right)$
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Find each of the following products:
$\left(\frac{4}{3} p q^{2}\right) \times\left(-\frac{1}{4} p^{2} r\right) \times\left(16 p^{2} q^{2} r^{2}\right)$
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Find each of the following products:
$(2.3 x y) \times(0.1 x) \times(0.16)$
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Express each of the following products as a monomials and verify the result in each case for $x=1$
$(3 x) \times(4 x) \times(-5 x)$
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Express each of the following products as a monomials and verify the result in each case for $x=1$
$\left(4 x^{2}\right) \times(-3 x) \times\left(\frac{4}{5} x^{3}\right)$
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Express each of the following products as a monomials and verify the result in each case for $x=1$
$\left(5 x^{4}\right) \times\left(x^{2}\right)^{3} \times(2 x)^{2}$
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Express each of the following products as a monomials and verify the result in each case for $x=1$
$\left(x^{2}\right)^{3} \times(2 x) \times(-4 x) \times(5)$
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Write down the product of $-8 x^{2} y^{6}$ and $-20 x y$. Verify the product for $x=2,5, y=1$.
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Evaluate $\left(3.2 x^{6} y^{3}\right) \times\left(2.1 x^{2} y^{2}\right)$ when $x=1$ and $y=0.5$
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Find the value of $\left(5 x^{6}\right) \times\left(-1.5 x^{2} y^{3}\right) \times\left(-12 x y^{2}\right)$ when $x=1, y=0.5$.
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Evaluate $\left(2.3 a^{5} b^{2}\right) \times\left(1.2 a^{2} b^{2}\right)$ when $a=1$ and $b=0.5$.
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Evaluate $\left(-8 x^{2} y^{6}\right) \times(-20 x y)$ for $x=2.5$ and $y=1$.
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Express each of the following products as a monomials and verify the result for $x \in l, y=2$ :
$\left(-x y^{3}\right) \times\left(y x^{3}\right) \times(x y)$
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Express each of the following products as a monomials and verify the result for $x \in l, y=2$ :
$\left(\frac{1}{8} x^{2} y^{4}\right) \times\left(\frac{1}{4} x^{4} y^{2}\right) \times(x y) \times 5$
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Express each of the following products as a monomials and verify the result for $x \in l, y=2$ :
$\left(\frac{2}{5} a^{2} b\right) \times\left(-15 b^{2} a c\right) \times\left(-\frac{1}{2} c^{2}\right)$
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Express each of the following products as a monomials and verify the result for $x \in l, y=2$ :
$\left(\frac{4}{9} a b c^{3}\right) \times\left(-\frac{27}{5} a^{3} b^{2}\right) \times\left(-8 b^{3} c\right)$
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Evaluate each of the following when $x=2, y=-1$.
$(2 x y) \times\left(\frac{x^{2} y}{4}\right) \times\left(x^{2}\right) \times\left(y^{2}\right) \quad$
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Evaluate each of the following when $x=2, y=-1$.
$\left(\frac{3}{5} x^{2} y\right) \times\left(-\frac{15}{4} x y^{2}\right) \times\left(\frac{7}{9} x^{2} y^{2}\right)$
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Ask Chatterbot AIExercise 6.4 - Algebraic Expressions and Identities | R.D. Sharma | Mathematics | Class 8
Find the following products: (1-15)
1. $2 a^{3}(3 a+5 b)$
2. $-11 a(3 a+2 b)$
3. $-5 a(7 a-2 b)$
4. $-11 y^{2}(3 y+7)$
5. $\frac{6 x}{5}\left(x^{3}+y^{3}\right)$
6. $x y\left(x^{3}-y^{3}\right)$
7. $0.1 y\left(0.1 x^{5}+0.1 y\right)$
8. $\left(-\frac{7}{4} a b^{2} c-\frac{6}{25} a^{2} c^{2}\right)\left(-50 a^{2} b^{2} c^{2}\right)$
9. $-\frac{8}{27} x y z\left(\frac{3}{2} x y z^{2}-\frac{9}{4} x y^{2} z^{3}\right)$
10. $-\frac{4}{27} x y z\left(\frac{9}{2} x^{2} y z-\frac{3}{4} x y z^{2}\right)$
11. $1.5 x\left(10 x^{2} y-100 x y^{2}\right)$
12. $4.1 x y(1.1 x-y)$
13. $250.5 x y\left(x z+\frac{y}{10}\right)$
14. $\frac{7}{5} x^{2} y\left(\frac{3}{5} x y^{2}+\frac{2}{5} x\right)$
15. $\frac{4}{3} a\left(a^{2}+b^{2}-3 c^{2}\right)$
Let's find the products for each given expression:
1. \(2a^3(3a + 5b)\)
Distribute \(2a^3\):
\[ 2a^3 \times 3a + 2a^3 \times 5b = 6a^4 + 10a^3b \]
So, the product is:
\[ \boxed{6a^4 + 10a^3b} \]
2. \(-11a(3a + 2b)\)
Distribute \(-11a\):
\[ -11a \times 3a + -11a \times 2b = -33a^2 - 22a b \]
So, the product is:
\[ \boxed{-33a^2 - 22ab} \]
3. \(-5a(7a - 2b)\)
Distribute \(-5a\):
\[ -5a \times 7a + -5a \times (-2b) = -35a^2 + 10ab \]
So, the product is:
\[ \boxed{-35a^2 + 10ab} \]
4. \(-11y^2(3y + 7)\)
Distribute \(-11y^2\):
\[ -11y^2 \times 3y + -11y^2 \times 7 = -33y^3 - 77y^2 \]
So, the product is:
\[ \boxed{-33y^3 - 77y^2} \]
5. \(\frac{6x}{5}(x^3 + y^3)\)
Distribute \(\frac{6x}{5}\):
\[ \frac{6x}{5} \times x^3 + \frac{6x}{5} \times y^3 = \frac{6x^4}{5} + \frac{6xy^3}{5} \]
So, the product is:
\[ \boxed{\frac{6x^4}{5} + \frac{6xy^3}{5}} \]
6. \(xy(x^3 - y^3)\)
Distribute \(xy\):
\[ xy \times x^3 - xy \times y^3 = x^4y - xy^4 \]
So, the product is:
\[ \boxed{x^4y - xy^4} \]
7. \(0.1y(0.1x^5 + 0.1y)\)
Distribute \(0.1y\):
\[ 0.1y \times 0.1x^5 + 0.1y \times 0.1y = 0.01x^5y + 0.01y^2 \]
So, the product is:
\[ \boxed{0.01x^5y + 0.01y^2} \]
8. \(\left(-\frac{7}{4}ab^2c - \frac{6}{25}a^2c^2\right)(-50a^2b^2c^2)\)
Distribute \(-50a^2b^2c^2\):
\[ \begin{aligned} &\left(-\frac{7}{4}ab^2c \right)(-50a^2b^2c^2) + \left(-\frac{6}{25}a^2c^2 \right)(-50a^2b^2c^2) \\ &= \frac{7}{4} \cdot 50 \cdot a^{2+1} \cdot b^{2+2} \cdot c^{1+2} + \frac{6}{25} \cdot 50 \cdot a^{2+2} \cdot b^2 \cdot c^{2+2} \\ &= \frac{7 \cdot 50}{4} \cdot a^3 \cdot b^4 \cdot c^3 + \frac{6 \cdot 50}{25} \cdot a^4 \cdot b^2 \cdot c^4 \\ &= 87.5a^3b^4c^3 + 12a^4b^2c^4 \end{aligned} \]
So, the product is:
\[ \boxed{87.5a^3b^4c^3 + 12a^4b^2c^4} \]
9. \(-\frac{8}{27}xyz\left(\frac{3}{2}xyz^2 - \frac{9}{4}xy^2z^3\right)\)
Distribute \(-\frac{8}{27}xyz\):
\[ -\frac{8}{27}xyz \times \frac{3}{2}xyz^2 - -\frac{8}{27}xyz \times \frac{9}{4}xy^2z^3 \]
Simplify each term:
\[ -\frac{8}{27} \cdot \frac{3}{2} \cdot x^2y^2 \cdot z^{2+1} - \left(-\frac{8}{27} \cdot \frac{9}{4} \cdot x^2y^3 \cdot z^{3+1}\right) \]
\[ -\frac{24}{54}x^2y^2z^3 + \frac{72}{108}x^2y^3z^4 \]
Simplify the fractions:
\[ -\frac{4}{9}x^2y^2z^3 + \frac{2}{3}x^2y^3z^4 \]
So, the product is:
\[ \boxed{-\frac{4}{9}x^2y^2z^3 + \frac{2}{3}x^2y^3z^4} \]
10. \(-\frac{4}{27}xyz\left(\frac{9}{2}x^2yz - \frac{3}{4}xyz^2\right)\)
Distribute \(-\frac{4}{27}xyz\):
\[ -\frac{4}{27}xyz \times \frac{9}{2}x^2yz - \left(-\frac{4}{27}xyz \times \frac{3}{4}xyz^2\right) \]
Simplify each term:
\[ -\frac{36}{54}x^3y^2z^2 + \frac{12}{108}x^2y^2z^{2+1} \]
Simplify the fractions:
\[ -\frac{2}{3}x^3y^2z^2 + \frac{1}{9}x^2y^2z^3 \]
So, the product is:
\[ \boxed{-\frac{2}{3}x^3y^2z^2 + \frac{1}{9}x^2y^2z^3} \]
11. \(1.5x(10x^2y - 100xy^2)\)
Distribute \(1.5x\):
\[ 1.5x \times 10x^2y + 1.5x \times -100xy^2 = 15x^3y - 150x^2y^2 \]
So, the product is:
\[ \boxed{15x^3y - 150x^2y^2} \]
12. \(4.1xy(1.1x - y)\)
Distribute \(4.1xy\):
\[ 4.1xy \times 1.1x + 4.1xy \times (-y) = 4.51x^2y - 4.1xy^2 \]
So, the product is:
\[ \boxed{4.51x^2y - 4.1xy^2} \]
13. \(250.5xy\left(xz + \frac{y}{10}\right)\)
Distribute \(250.5xy\):
\[ 250.5xy \times xz + 250.5xy \times \left(\frac{y}{10}\right) \]
\[ = 250.5x^2yz + 250.5xy \times \frac{y}{10} = 250.5x^2yz + 25.05xy^2 \]
So, the product is:
\[ \boxed{250.5x^2yz + 25.05xy^2} \]
14. \(\frac{7}{5}x^2y\left(\frac{3}{5}xy^2 + \frac{2}{5}x\right)\)
Distribute \(\frac{7}{5}x^2y\):
\[ \frac{7}{5}x^2y \times \frac{3}{5}xy^2 + \frac{7}{5}x^2y \times \frac{2}{5}x \]
\[ = \frac{21}{25}x^3y^3 + \frac{14}{25}x^3y \]
So, the product is:
\[ \boxed{\frac{21}{25}x^3y^3 + \frac{14}{25}x^3y} \]
15. \(\frac{4}{3}a\left(a^2 + b^2 - 3c^2\right)\)
Distribute \(\frac{4}{3}a\):
\[ \frac{4}{3}a \times a^2 + \frac{4}{3}a \times b^2 - \frac{4}{3}a \times 3c^2 = \frac{4}{3}a^3 + \frac{4}{3}ab^2 - 4ac^2 \]
So, the product is:
\[ \boxed{\frac{4}{3}a^3 + \frac{4}{3}ab^2 - 4ac^2} \]
Find the product $24 x^{2}(1-2 x)$ and evaluate its value for $x=3$.
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Find the product $-3 y\left(x y+y^{2}\right)$ and find its value for $x=4$ and $y=5$.
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Multiply $-\frac{3}{2} x^{2} y^{3}$ by $(2 x-y)$ and verify the answer for $x=1$ and $y=2$.
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Multiply the monomial by the binomial and find the value of each for $x=-1, y=0.25$ and $z=0.05$ :
(i) $15 y^{2}(2-3 x)$
(ii) $-3 x\left(y^{2}+z^{2}\right)$
(iii) $z^{2}(x-y)$
(iv) $x z\left(x^{2}+y^{2}\right)$
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Simplify:
(i) $2 x^{2}\left(x^{3}-x\right)-3 x\left(x^{4}+2 x\right)-2\left(x^{4}-3 x^{2}\right)$
(ii) $x^{3} y\left(x^{2}-2 x\right)+2 x y\left(x^{3}-x^{4}\right)$
(iii) $3 a^{2}+2(a+2)-3 a(2 a+1)$
(iv) $x(x+4)+3 x\left(2 x^{2}-1\right)+4 x^{2}+4$
(v) $a(b-c)-b(c-a)-c(a-b)$
(vi) $a(b-c)+b(c-a)+c(a-b)$
(vii) $4 a b(a-b)-6 a^{2}\left(b-b^{2}\right)-3 b^{2}\left(2 a^{2}-a\right)+2 a b(b-a)$
(viii) $x^{2}\left(x^{2}+1\right)-x^{3}(x+1)-x\left(x^{3}-x\right)$
(ix) $2 a^{2}+3 a\left(1-2 a^{3}\right)+a(a+1)$
(x) $a^{2}(2 a-1)+3 a+a^{3}-8$
(xi) $\frac{3}{2} x^{2}\left(x^{2}-1\right)+\frac{1}{4} x^{2}\left(x^{2}+x\right)-\frac{3}{4} x\left(x^{3}-1\right)$
(xii) $a^{2} b\left(a-b^{2}\right)+a b^{2}\left(4 a b-2 a^{2}\right)-a^{3} b(1-2 b)$
(xiii) $a^{2} b\left(a^{3}-a+1\right)-a b\left(a^{4}-2 a^{2}+2 a\right)-b\left(a^{3}-a^{2}-1\right)$
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Ask Chatterbot AIExercise 6.5 - Algebraic Expressions and Identities | R.D. Sharma | Mathematics | Class 8
Multiply:
$(5 x+3)$ by $(7 x+2)$
To multiply the binomials ( (5x + 3) ) and ( (7x + 2) ), we'll use the distributive property:
$$ (5x + 3)(7x + 2) = 5x \cdot (7x + 2) + 3 \cdot (7x + 2) $$
This expands to:
$$ 5x \cdot 7x + 5x \cdot 2 + 3 \cdot 7x + 3 \cdot 2 $$
Further simplifying:
$$ 35x^2 + 10x + 21x + 6 $$
Combining like terms:
$$ 35x^2 + 31x + 6 $$
Thus, the product of the binomials is:
$$ 35x^2 + 31x + 6 $$
Multiply:
$(7 x+y)$ by $(x+5 y)$
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Multiply:
$(2 x+8)$ by $(x-3)$
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Multiply:
$(a-1)$ by $\left(0.1 a^{2}+3\right)$
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Multiply:
$\left(3 x^{2}+y^{2}\right)$ by $\left(2 x^{2}+3 y^{2}\right)$
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Multiply:
$\left(\frac{3}{5} x+\frac{1}{2} y\right)$ by $\left(\frac{5}{6} x+4 y\right)$
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Multiply:
$\left(x^{6}-y^{6}\right)$ by $\left(x^{2}+y^{2}\right)$
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Multiply:
$\left(x^{2}+y^{2}\right)$ by $(3 a+2 b)$
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Multiply:
$[-3 d+(-7 f)]$ by $(5 d+f)$
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Multiply:
$(0.8 a-0.5 b)$ by $(1.5 a-3 b)$
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Multiply:
$\left(2 x^{2} y^{2}-5 x y^{2}\right)$ by $\left(x^{2}-y^{2}\right)$
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Multiply:
$\left(\frac{x}{7}+\frac{x^{2}}{2}\right)$ by $\left(\frac{2}{5}+\frac{9 x}{4}\right)$
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Multiply:
$\left(-\frac{a}{7}+\frac{a^{2}}{9}\right)$ by $\left(\frac{b}{2}-\frac{b^{2}}{3}\right)$
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Multiply:
$\left(3 x^{2} y-5 x y^{2}\right)$ by $\left(\frac{1}{5} x^{2}+\frac{1}{3} y^{2}\right)$
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Multiply:
$\left(2 x^{2}-1\right)$ by $\left(4 x^{3}+5 x^{2}\right)$
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Multiply:
$\left(2 x y+3 y^{2}\right)\left(3 y^{2}-2\right)$
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Find the following products and verify the result for $x=-1, y=-2$ :
$(3 x-5 y)(x+y)$
$\left(x^{2} y-1\right)\left(3-2 x^{2} y\right)$
$\left(\frac{1}{3} x-\frac{y^{2}}{5}\right)\left(\frac{1}{3} x+\frac{y^{2}}{5}\right)$
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Simplify:
$x^{2}(x+2 y)(x-3 y)$
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Simplify:
$\left(x^{2}-2 y^{2}\right)(x+4 y) x^{2} y^{2}$
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Simplify:
$a^{2} b^{2}(a+2 b)(3 a+b)$
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Simplify:
$x^{2}(x-y) y^{2}(x+2 y)$
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Simplify:
$\left(x^{3}-2 x^{2}+5 x-7\right)(2 x-3)$
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Simplify:
$(5 x+3)(x-1)(3 x-2)$
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Simplify:
$(5-x)(6-5 x)(2-x)$
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Simplify:
$\left(2 x^{2}+3 x-5\right)\left(3 x^{2}-5 x+4\right)$
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Simplify:
$(3 x-2)(2 x-3)+(5 x-3)(x+1)$
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Simplify:
$(5 x-3)(x+2)-(2 x+5)(4 x-3)$
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Simplify:
$(3 x+2 y)(4 x+3 y)-(2 x-y)(7 x-3 y)$
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Simplify:
$\left(x^{2}-3 x+2\right)(5 x-2)-\left(3 x^{2}+4 x-5\right)(2 x-1)$
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Simplify:
$\left(x^{3}-2 x^{2}+3 x-4\right)(x-1)-(2 x-3)\left(x^{2}-x+1\right)$
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Ask Chatterbot AIExtra Questions - Algebraic Expressions and Identities | R.D. Sharma | Mathematics | Class 8
Select the statements that are true.
A $(-x) y = x(-y)$
B $(-x) y = -(x y)$
C $x(-y) = -(x y)$
D $x(-y) = (x y)$
Selected true statements:
A: $(-x)y = x(-y)$
B: $(-x)y = -(xy)$
C: $x(-y) = -(xy)$
Explanation:
The multiplication involving negatives can be understood by considering how signs interact with each other:
-
The expression $(-x)y$ implies multiplying a negative value $-x$ with a positive value $y$. According to sign multiplication rules, a negative times a positive is negative, thus $(-x)y = -(xy)$.
-
Similarly, $x(-y)$ translates to multiplying a positive value $x$ with a negative value $-y$. Following the same rule, a positive times a negative yields a negative, leading to $x(-y) = -(xy)$.
-
Both expressions $(-x)y$ and $x(-y)$ essentially end up implying the same arithmetic operation: multiply $x$ and $y$ and then invert the sign. Therefore, we deduce $(-x)y = x(-y)$.
This reasoning demonstrates that statements A, B, and C are true, while statement D is false, as it incorrectly equates $x(-y)$ to $(xy)$, neglecting the sign change.
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